Theorem efrirr 5611

Description: A well-founded class does not belong to itself. (Contributed by NM, 18-Apr-1994.) (Revised by Mario Carneiro, 22-Jun-2015.)

Assertion

Ref	Expression
efrirr	⊢ ( E Fr 𝐴 → ¬ 𝐴 ∈ 𝐴)

Proof of Theorem efrirr

Step	Hyp	Ref	Expression
1		frirr 5607	. . 3 ⊢ (( E Fr 𝐴 ∧ 𝐴 ∈ 𝐴) → ¬ 𝐴 E 𝐴)
2		epelg 5532	. . . 4 ⊢ (𝐴 ∈ 𝐴 → (𝐴 E 𝐴 ↔ 𝐴 ∈ 𝐴))
3	2	adantl 481	. . 3 ⊢ (( E Fr 𝐴 ∧ 𝐴 ∈ 𝐴) → (𝐴 E 𝐴 ↔ 𝐴 ∈ 𝐴))
4	1, 3	mtbid 324	. 2 ⊢ (( E Fr 𝐴 ∧ 𝐴 ∈ 𝐴) → ¬ 𝐴 ∈ 𝐴)
5	4	pm2.01da 799	1 ⊢ ( E Fr 𝐴 → ¬ 𝐴 ∈ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∈ wcel 2114   class class class wbr 5085   E cep 5530   Fr wfr 5581

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708  ax-sep 5231  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-br 5086  df-opab 5148  df-eprel 5531  df-fr 5584

This theorem is referenced by:  tz7.2  5614  ordirr  6341  elirrvALT  9526